Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena.^{[1]}^{[2]}^{[3]} Two of the most highly abstract areas of modern mathematics are category theory and model theory.
Description
Many areas of mathematics began with the study of real world problems, before the underlying rules and concepts were identified and defined as abstract structures. For example, geometry has its origins in the calculation of distances and areas in the real world; algebra started with methods of solving problems in arithmetic.
Abstraction is an ongoing process in mathematics and the historical development of many mathematical topics exhibits a progression from the concrete to the abstract. Take the historical development of geometry as an example; the first steps in the abstraction of geometry were made by the ancient Greeks, with Euclid's Elements being the earliest extant documentation of the axioms of plane geometry—though Proclus tells of an earlier axiomatisation by Hippocrates of Chios.^{[4]} In the 17th century Descartes introduced Cartesian coordinates which allowed the development of analytic geometry. Further steps in abstraction were taken by Lobachevsky, Bolyai, Riemann, and Gauss who generalised the concepts of geometry to develop nonEuclidean geometries. Later in the 19th century mathematicians generalised geometry even further, developing such areas as geometry in n dimensions, projective geometry, affine geometry and finite geometry. Finally Felix Klein's "Erlangen program" identified the underlying theme of all of these geometries, defining each of them as the study of properties invariant under a given group of symmetries. This level of abstraction revealed connections between geometry and abstract algebra.
The advantages of abstraction are :

It reveals deep connections between different areas of mathematics.

Known results in one area can suggest conjectures in a related area.

Techniques and methods from one area can be applied to prove results in a related area.
One disadvantage of abstraction is that highly abstract concepts can be difficult to learn.^{[5]} A degree of mathematical maturity and experience may be needed for conceptual assimilation of abstractions. One of the underlying principles of the Montessori approach to mathematics education is encouraging children to move from concrete examples to abstract thinking.^{[6]}
Bertrand Russell, in The Scientific Outlook (1931), writes that "Ordinary language is totally unsuited for expressing what physics really asserts, since the words of everyday life are not sufficiently abstract. Only mathematics and mathematical logic can say as little as the physicist means to say."
See also
References

^ Bertrand Russell, in The Principles of Mathematics Volume 1 (pg 219), refers to "the principle of abstraction".

^ Robert B. Ash. A Primer of Abstract Mathematics. Cambridge University Press, Jan 1, 1998

^ The New American Encyclopedic Dictionary. Edited by Edward Thomas Roe, Le Roy Hooker, Thomas W. Handford. Pg 34

^ Proclus' Summary

^ "... introducing pupils to abstract mathematics is not an easy task and requires a longterm effort that must take into account the variety of the contexts in which mathematics is used", P.L. Ferrari, Abstraction in Mathematics, Phil. Trans. R. Soc. Lond. B 29 July 2003 vol. 358 no. 1435 12251230

^ Montessori Philosophy: Moving from Concrete to Abstract, North American Montessori Center
Further reading

Bajnok, Béla (2013). An Invitation to Abstract Mathematics. Springer.
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.